Jmat.Real.erfi, branch x less than or equal to 0.5

Percentage Accurate: 99.8% → 99.9%

Time: 8.1s

Alternatives: 7

Speedup: 3.7×

• 28.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t_0\right) + \frac{1}{5} \cdot t_1\right) + \frac{1}{21} \cdot \left(\left(t_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
      (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * fabs(x)) * fabs(x))))));
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * Math.abs(x)) * Math.abs(x))))));
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * math.fabs(x)) * math.fabs(x))))))

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_1 * abs(x)) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * abs(x)) * abs(x))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t_0\right) + \frac{1}{5} \cdot t_1\right) + \frac{1}{21} \cdot \left(\left(t_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
      (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * fabs(x)) * fabs(x))))));
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * Math.abs(x)) * Math.abs(x))))));
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * math.fabs(x)) * math.fabs(x))))))

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_1 * abs(x)) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * abs(x)) * abs(x))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Alternative 1: 99.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \left|\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\left(\left(e^{\mathsf{log1p}\left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)} + -1\right) + 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (* x (sqrt (/ 1.0 PI)))
   (+
    (+ (+ (exp (log1p (* 0.6666666666666666 (* x x)))) -1.0) 2.0)
    (+ (* 0.2 (pow x 4.0)) (* 0.047619047619047616 (pow x 6.0)))))))

C

double code(double x) {
	return fabs(((x * sqrt((1.0 / ((double) M_PI)))) * (((exp(log1p((0.6666666666666666 * (x * x)))) + -1.0) + 2.0) + ((0.2 * pow(x, 4.0)) + (0.047619047619047616 * pow(x, 6.0))))));
}

Java

public static double code(double x) {
	return Math.abs(((x * Math.sqrt((1.0 / Math.PI))) * (((Math.exp(Math.log1p((0.6666666666666666 * (x * x)))) + -1.0) + 2.0) + ((0.2 * Math.pow(x, 4.0)) + (0.047619047619047616 * Math.pow(x, 6.0))))));
}

Python

def code(x):
	return math.fabs(((x * math.sqrt((1.0 / math.pi))) * (((math.exp(math.log1p((0.6666666666666666 * (x * x)))) + -1.0) + 2.0) + ((0.2 * math.pow(x, 4.0)) + (0.047619047619047616 * math.pow(x, 6.0))))))

Julia

function code(x)
	return abs(Float64(Float64(x * sqrt(Float64(1.0 / pi))) * Float64(Float64(Float64(exp(log1p(Float64(0.6666666666666666 * Float64(x * x)))) + -1.0) + 2.0) + Float64(Float64(0.2 * (x ^ 4.0)) + Float64(0.047619047619047616 * (x ^ 6.0))))))
end

Wolfram

code[x_] := N[Abs[N[(N[(x * N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Exp[N[Log[1 + N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $MachinePrecision] + N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right|} \]

4. Taylor expanded in x around 0 99.9%

   \[\leadsto \left|\color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]
5. Step-by-step derivation

   1. *-commutative 99.9%

      \[\leadsto \left|\color{blue}{\left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   2. unpow1 99.9%

      \[\leadsto \left|\left(\left|\color{blue}{{x}^{1}}\right| \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   3. sqr-pow 31.0%

      \[\leadsto \left|\left(\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   4. fabs-sqr 31.0%

      \[\leadsto \left|\left(\color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)} \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   5. sqr-pow 99.9%

      \[\leadsto \left|\left(\color{blue}{{x}^{1}} \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   6. unpow1 99.9%

      \[\leadsto \left|\left(\color{blue}{x} \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

6. Simplified 99.9%

   \[\leadsto \left|\color{blue}{\left(x \cdot \sqrt{\frac{1}{\pi}}\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]
7. Step-by-step derivation

   1. fma-udef 99.9%

      \[\leadsto \left|\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \color{blue}{\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)}\right)\right| \]

8. Applied egg-rr 99.9%

   \[\leadsto \left|\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \color{blue}{\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)}\right)\right| \]
9. Step-by-step derivation

   1. fma-udef 99.9%

      \[\leadsto \left|\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\color{blue}{\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right)} + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

10. Applied egg-rr 99.9%

    \[\leadsto \left|\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\color{blue}{\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right)} + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]
11. Step-by-step derivation

    1. associate-*r* 99.9%

       \[\leadsto \left|\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\left(\color{blue}{\left(0.6666666666666666 \cdot x\right) \cdot x} + 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    2. *-commutative 99.9%

       \[\leadsto \left|\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\left(\color{blue}{x \cdot \left(0.6666666666666666 \cdot x\right)} + 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    3. expm1-log1p-u 99.9%

       \[\leadsto \left|\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(0.6666666666666666 \cdot x\right)\right)\right)} + 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    4. expm1-udef 99.9%

       \[\leadsto \left|\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\left(\color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot \left(0.6666666666666666 \cdot x\right)\right)} - 1\right)} + 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    5. *-commutative 99.9%

       \[\leadsto \left|\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\left(\left(e^{\mathsf{log1p}\left(\color{blue}{\left(0.6666666666666666 \cdot x\right) \cdot x}\right)} - 1\right) + 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    6. associate-*r* 99.9%

       \[\leadsto \left|\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\left(\left(e^{\mathsf{log1p}\left(\color{blue}{0.6666666666666666 \cdot \left(x \cdot x\right)}\right)} - 1\right) + 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

12. Applied egg-rr 99.9%

    \[\leadsto \left|\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\left(\color{blue}{\left(e^{\mathsf{log1p}\left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)} - 1\right)} + 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

13. Final simplification 99.9%

    \[\leadsto \left|\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\left(\left(e^{\mathsf{log1p}\left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)} + -1\right) + 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

Alternative 2: 99.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + \left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (* x (pow PI -0.5))
   (+
    (+ (* 0.2 (pow x 4.0)) (* 0.047619047619047616 (pow x 6.0)))
    (+ (* 0.6666666666666666 (* x x)) 2.0)))))

C

double code(double x) {
	return fabs(((x * pow(((double) M_PI), -0.5)) * (((0.2 * pow(x, 4.0)) + (0.047619047619047616 * pow(x, 6.0))) + ((0.6666666666666666 * (x * x)) + 2.0))));
}

Java

public static double code(double x) {
	return Math.abs(((x * Math.pow(Math.PI, -0.5)) * (((0.2 * Math.pow(x, 4.0)) + (0.047619047619047616 * Math.pow(x, 6.0))) + ((0.6666666666666666 * (x * x)) + 2.0))));
}

Python

def code(x):
	return math.fabs(((x * math.pow(math.pi, -0.5)) * (((0.2 * math.pow(x, 4.0)) + (0.047619047619047616 * math.pow(x, 6.0))) + ((0.6666666666666666 * (x * x)) + 2.0))))

Julia

function code(x)
	return abs(Float64(Float64(x * (pi ^ -0.5)) * Float64(Float64(Float64(0.2 * (x ^ 4.0)) + Float64(0.047619047619047616 * (x ^ 6.0))) + Float64(Float64(0.6666666666666666 * Float64(x * x)) + 2.0))))
end

MATLAB

function tmp = code(x)
	tmp = abs(((x * (pi ^ -0.5)) * (((0.2 * (x ^ 4.0)) + (0.047619047619047616 * (x ^ 6.0))) + ((0.6666666666666666 * (x * x)) + 2.0))));
end

Wolfram

code[x_] := N[Abs[N[(N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right|} \]

4. Taylor expanded in x around 0 99.9%

   \[\leadsto \left|\color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]
5. Step-by-step derivation

   1. *-commutative 99.9%

      \[\leadsto \left|\color{blue}{\left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   2. unpow1 99.9%

      \[\leadsto \left|\left(\left|\color{blue}{{x}^{1}}\right| \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   3. sqr-pow 31.0%

      \[\leadsto \left|\left(\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   4. fabs-sqr 31.0%

      \[\leadsto \left|\left(\color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)} \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   5. sqr-pow 99.9%

      \[\leadsto \left|\left(\color{blue}{{x}^{1}} \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   6. unpow1 99.9%

      \[\leadsto \left|\left(\color{blue}{x} \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

6. Simplified 99.9%

   \[\leadsto \left|\color{blue}{\left(x \cdot \sqrt{\frac{1}{\pi}}\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]
7. Step-by-step derivation

   1. fma-udef 99.9%

      \[\leadsto \left|\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \color{blue}{\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)}\right)\right| \]

8. Applied egg-rr 99.9%

   \[\leadsto \left|\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \color{blue}{\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)}\right)\right| \]
9. Step-by-step derivation

   1. fma-udef 99.9%

      \[\leadsto \left|\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\color{blue}{\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right)} + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

10. Applied egg-rr 99.9%

    \[\leadsto \left|\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\color{blue}{\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right)} + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]
11. Step-by-step derivation

    1. expm1-log1p-u 66.7%

       \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \cdot \left(\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    2. expm1-udef 6.1%

       \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot \sqrt{\frac{1}{\pi}}\right)} - 1\right)} \cdot \left(\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    3. pow1/2 6.1%

       \[\leadsto \left|\left(e^{\mathsf{log1p}\left(x \cdot \color{blue}{{\left(\frac{1}{\pi}\right)}^{0.5}}\right)} - 1\right) \cdot \left(\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    4. inv-pow 6.1%

       \[\leadsto \left|\left(e^{\mathsf{log1p}\left(x \cdot {\color{blue}{\left({\pi}^{-1}\right)}}^{0.5}\right)} - 1\right) \cdot \left(\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    5. pow-pow 6.1%

       \[\leadsto \left|\left(e^{\mathsf{log1p}\left(x \cdot \color{blue}{{\pi}^{\left(-1 \cdot 0.5\right)}}\right)} - 1\right) \cdot \left(\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    6. metadata-eval 6.1%

       \[\leadsto \left|\left(e^{\mathsf{log1p}\left(x \cdot {\pi}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot \left(\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

12. Applied egg-rr 6.1%

    \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)} - 1\right)} \cdot \left(\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]
13. Step-by-step derivation

    1. expm1-def 66.7%

       \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)\right)} \cdot \left(\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    2. expm1-log1p 99.9%

       \[\leadsto \left|\color{blue}{\left(x \cdot {\pi}^{-0.5}\right)} \cdot \left(\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

14. Simplified 99.9%

    \[\leadsto \left|\color{blue}{\left(x \cdot {\pi}^{-0.5}\right)} \cdot \left(\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

15. Final simplification 99.9%

    \[\leadsto \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + \left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right)\right)\right| \]

Alternative 3: 99.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \left|x \cdot \left({\pi}^{-0.5} \cdot \left(0.2 \cdot {x}^{4} + \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   x
   (*
    (pow PI -0.5)
    (+ (* 0.2 (pow x 4.0)) (+ 2.0 (* 0.047619047619047616 (pow x 6.0))))))))

C

double code(double x) {
	return fabs((x * (pow(((double) M_PI), -0.5) * ((0.2 * pow(x, 4.0)) + (2.0 + (0.047619047619047616 * pow(x, 6.0)))))));
}

Java

public static double code(double x) {
	return Math.abs((x * (Math.pow(Math.PI, -0.5) * ((0.2 * Math.pow(x, 4.0)) + (2.0 + (0.047619047619047616 * Math.pow(x, 6.0)))))));
}

Python

def code(x):
	return math.fabs((x * (math.pow(math.pi, -0.5) * ((0.2 * math.pow(x, 4.0)) + (2.0 + (0.047619047619047616 * math.pow(x, 6.0)))))))

Julia

function code(x)
	return abs(Float64(x * Float64((pi ^ -0.5) * Float64(Float64(0.2 * (x ^ 4.0)) + Float64(2.0 + Float64(0.047619047619047616 * (x ^ 6.0)))))))
end

MATLAB

function tmp = code(x)
	tmp = abs((x * ((pi ^ -0.5) * ((0.2 * (x ^ 4.0)) + (2.0 + (0.047619047619047616 * (x ^ 6.0)))))));
end

Wolfram

code[x_] := N[Abs[N[(x * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{\left|\frac{\left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot 0.6666666666666666, x, 2\right) + \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{4}, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6}\right)\right)}{\sqrt{\pi}}\right|} \]

4. Taylor expanded in x around 0 99.0%

   \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right)\right)}\right| \]
5. Step-by-step derivation

   1. *-commutative 99.0%

      \[\leadsto \left|\color{blue}{\left(\left|x\right| \cdot \left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right)\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

   2. associate-*l* 99.1%

      \[\leadsto \left|\color{blue}{\left|x\right| \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

   3. unpow1 99.1%

      \[\leadsto \left|\left|\color{blue}{{x}^{1}}\right| \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

   4. sqr-pow 30.8%

      \[\leadsto \left|\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

   5. fabs-sqr 30.8%

      \[\leadsto \left|\color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

   6. sqr-pow 99.1%

      \[\leadsto \left|\color{blue}{{x}^{1}} \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

   7. unpow1 99.1%

      \[\leadsto \left|\color{blue}{x} \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

6. Simplified 99.1%

   \[\leadsto \left|\color{blue}{x \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

7. Taylor expanded in x around 0 99.1%

   \[\leadsto \left|x \cdot \left(\color{blue}{\left(2 + \left(0.047619047619047616 \cdot {x}^{6} + 0.2 \cdot {x}^{4}\right)\right)} \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]
8. Step-by-step derivation

   1. *-commutative 99.1%

      \[\leadsto \left|x \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + 0.2 \cdot {x}^{4}\right)\right)\right)}\right| \]

   2. associate-+r+ 99.1%

      \[\leadsto \left|x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(\left(2 + 0.047619047619047616 \cdot {x}^{6}\right) + 0.2 \cdot {x}^{4}\right)}\right)\right| \]

   3. distribute-lft-in 99.1%

      \[\leadsto \left|x \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{4}\right)\right)}\right| \]

   4. inv-pow 99.1%

      \[\leadsto \left|x \cdot \left(\sqrt{\color{blue}{{\pi}^{-1}}} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{4}\right)\right)\right| \]

   5. sqrt-pow1 99.1%

      \[\leadsto \left|x \cdot \left(\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{4}\right)\right)\right| \]

   6. metadata-eval 99.1%

      \[\leadsto \left|x \cdot \left({\pi}^{\color{blue}{-0.5}} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{4}\right)\right)\right| \]

   7. inv-pow 99.1%

      \[\leadsto \left|x \cdot \left({\pi}^{-0.5} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right) + \sqrt{\color{blue}{{\pi}^{-1}}} \cdot \left(0.2 \cdot {x}^{4}\right)\right)\right| \]

   8. sqrt-pow1 99.1%

      \[\leadsto \left|x \cdot \left({\pi}^{-0.5} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right) + \color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left(0.2 \cdot {x}^{4}\right)\right)\right| \]

   9. metadata-eval 99.1%

      \[\leadsto \left|x \cdot \left({\pi}^{-0.5} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right) + {\pi}^{\color{blue}{-0.5}} \cdot \left(0.2 \cdot {x}^{4}\right)\right)\right| \]

9. Applied egg-rr 99.1%

   \[\leadsto \left|x \cdot \color{blue}{\left({\pi}^{-0.5} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right) + {\pi}^{-0.5} \cdot \left(0.2 \cdot {x}^{4}\right)\right)}\right| \]
10. Step-by-step derivation

    1. distribute-lft-out 99.1%

       \[\leadsto \left|x \cdot \color{blue}{\left({\pi}^{-0.5} \cdot \left(\left(2 + 0.047619047619047616 \cdot {x}^{6}\right) + 0.2 \cdot {x}^{4}\right)\right)}\right| \]

11. Simplified 99.1%

    \[\leadsto \left|x \cdot \color{blue}{\left({\pi}^{-0.5} \cdot \left(\left(2 + 0.047619047619047616 \cdot {x}^{6}\right) + 0.2 \cdot {x}^{4}\right)\right)}\right| \]

12. Final simplification 99.1%

    \[\leadsto \left|x \cdot \left({\pi}^{-0.5} \cdot \left(0.2 \cdot {x}^{4} + \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right)\right| \]

Alternative 4: 89.1% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   (fabs (* (sqrt (/ 1.0 PI)) (* x (fma 0.6666666666666666 (* x x) 2.0))))
   (fabs (* (pow x 7.0) (/ 0.047619047619047616 (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * (x * fma(0.6666666666666666, (x * x), 2.0))));
	} else {
		tmp = fabs((pow(x, 7.0) * (0.047619047619047616 / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2)
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(x * fma(0.6666666666666666, Float64(x * x), 2.0))));
	else
		tmp = abs(Float64((x ^ 7.0) * Float64(0.047619047619047616 / sqrt(pi))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x * N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Power[x, 7.0], $MachinePrecision] * N[(0.047619047619047616 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(0.047619047619047616, \left({\left(\left|x\right|\right)}^{3} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right), \mathsf{fma}\left(2, \left|x\right|, \mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, {\left(\left|x\right|\right)}^{3} \cdot \left(\left(x \cdot x\right) \cdot 0.2\right)\right)\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around 0 90.6%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)}\right| \]
   5. Step-by-step derivation

      1. unpow3 90.6%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot \color{blue}{\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)} + 2 \cdot \left|x\right|\right)\right| \]

      2. sqr-abs 90.6%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left|x\right|\right) + 2 \cdot \left|x\right|\right)\right| \]

      3. unpow2 90.6%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot \left(\color{blue}{{x}^{2}} \cdot \left|x\right|\right) + 2 \cdot \left|x\right|\right)\right| \]

      4. associate-*r* 90.6%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\left(0.6666666666666666 \cdot {x}^{2}\right) \cdot \left|x\right|} + 2 \cdot \left|x\right|\right)\right| \]

      5. distribute-rgt-in 90.6%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(\left|x\right| \cdot \left(0.6666666666666666 \cdot {x}^{2} + 2\right)\right)}\right| \]

      6. unpow1 90.6%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|\color{blue}{{x}^{1}}\right| \cdot \left(0.6666666666666666 \cdot {x}^{2} + 2\right)\right)\right| \]

      7. sqr-pow 31.0%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(0.6666666666666666 \cdot {x}^{2} + 2\right)\right)\right| \]

      8. fabs-sqr 31.0%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(0.6666666666666666 \cdot {x}^{2} + 2\right)\right)\right| \]

      9. sqr-pow 90.6%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{{x}^{1}} \cdot \left(0.6666666666666666 \cdot {x}^{2} + 2\right)\right)\right| \]

      10. unpow1 90.6%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{x} \cdot \left(0.6666666666666666 \cdot {x}^{2} + 2\right)\right)\right| \]

      11. fma-def 90.6%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)}\right)\right| \]

      12. unpow2 90.6%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \mathsf{fma}\left(0.6666666666666666, \color{blue}{x \cdot x}, 2\right)\right)\right| \]

   6. Simplified 90.6%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)}\right| \]

   if 2.2000000000000002 < x

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left|\frac{\left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot 0.6666666666666666, x, 2\right) + \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{4}, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6}\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around 0 99.0%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right)\right)}\right| \]
   5. Step-by-step derivation

      1. *-commutative 99.0%

         \[\leadsto \left|\color{blue}{\left(\left|x\right| \cdot \left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right)\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

      2. associate-*l* 99.1%

         \[\leadsto \left|\color{blue}{\left|x\right| \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

      3. unpow1 99.1%

         \[\leadsto \left|\left|\color{blue}{{x}^{1}}\right| \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      4. sqr-pow 30.8%

         \[\leadsto \left|\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      5. fabs-sqr 30.8%

         \[\leadsto \left|\color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      6. sqr-pow 99.1%

         \[\leadsto \left|\color{blue}{{x}^{1}} \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      7. unpow1 99.1%

         \[\leadsto \left|\color{blue}{x} \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

   6. Simplified 99.1%

      \[\leadsto \left|\color{blue}{x \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

   7. Taylor expanded in x around inf 36.6%

      \[\leadsto \left|x \cdot \color{blue}{\left(0.047619047619047616 \cdot \left({x}^{6} \cdot \sqrt{\frac{1}{\pi}}\right)\right)}\right| \]
   8. Step-by-step derivation

      1. associate-*r* 36.7%

         \[\leadsto \left|x \cdot \color{blue}{\left(\left(0.047619047619047616 \cdot {x}^{6}\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

   9. Simplified 36.7%

      \[\leadsto \left|x \cdot \color{blue}{\left(\left(0.047619047619047616 \cdot {x}^{6}\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   10. Step-by-step derivation

       1. expm1-log1p-u 3.8%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(\left(0.047619047619047616 \cdot {x}^{6}\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)}\right| \]

       2. expm1-udef 3.5%

          \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(x \cdot \left(\left(0.047619047619047616 \cdot {x}^{6}\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)} - 1}\right| \]

       3. associate-*l* 3.5%

          \[\leadsto \left|e^{\mathsf{log1p}\left(x \cdot \color{blue}{\left(0.047619047619047616 \cdot \left({x}^{6} \cdot \sqrt{\frac{1}{\pi}}\right)\right)}\right)} - 1\right| \]

       4. sqrt-div 3.5%

          \[\leadsto \left|e^{\mathsf{log1p}\left(x \cdot \left(0.047619047619047616 \cdot \left({x}^{6} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right)\right)\right)} - 1\right| \]

       5. metadata-eval 3.5%

          \[\leadsto \left|e^{\mathsf{log1p}\left(x \cdot \left(0.047619047619047616 \cdot \left({x}^{6} \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right)\right)\right)} - 1\right| \]

       6. un-div-inv 3.5%

          \[\leadsto \left|e^{\mathsf{log1p}\left(x \cdot \left(0.047619047619047616 \cdot \color{blue}{\frac{{x}^{6}}{\sqrt{\pi}}}\right)\right)} - 1\right| \]

   11. Applied egg-rr 3.5%

       \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(x \cdot \left(0.047619047619047616 \cdot \frac{{x}^{6}}{\sqrt{\pi}}\right)\right)} - 1}\right| \]
   12. Step-by-step derivation

       1. expm1-def 3.8%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(0.047619047619047616 \cdot \frac{{x}^{6}}{\sqrt{\pi}}\right)\right)\right)}\right| \]

       2. expm1-log1p 36.7%

          \[\leadsto \left|\color{blue}{x \cdot \left(0.047619047619047616 \cdot \frac{{x}^{6}}{\sqrt{\pi}}\right)}\right| \]

       3. associate-*r/ 36.7%

          \[\leadsto \left|x \cdot \color{blue}{\frac{0.047619047619047616 \cdot {x}^{6}}{\sqrt{\pi}}}\right| \]

       4. associate-*r/ 36.7%

          \[\leadsto \left|\color{blue}{\frac{x \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}}\right| \]

       5. *-commutative 36.7%

          \[\leadsto \left|\frac{\color{blue}{\left(0.047619047619047616 \cdot {x}^{6}\right) \cdot x}}{\sqrt{\pi}}\right| \]

       6. associate-*l* 36.7%

          \[\leadsto \left|\frac{\color{blue}{0.047619047619047616 \cdot \left({x}^{6} \cdot x\right)}}{\sqrt{\pi}}\right| \]

       7. pow-plus 36.7%

          \[\leadsto \left|\frac{0.047619047619047616 \cdot \color{blue}{{x}^{\left(6 + 1\right)}}}{\sqrt{\pi}}\right| \]

       8. metadata-eval 36.7%

          \[\leadsto \left|\frac{0.047619047619047616 \cdot {x}^{\color{blue}{7}}}{\sqrt{\pi}}\right| \]

   13. Simplified 36.7%

       \[\leadsto \left|\color{blue}{\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}}\right| \]
   14. Step-by-step derivation

       1. expm1-log1p-u 3.8%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right)\right)}\right| \]

       2. expm1-udef 3.5%

          \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right)} - 1}\right| \]

       3. associate-/l* 3.5%

          \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{0.047619047619047616}{\frac{\sqrt{\pi}}{{x}^{7}}}}\right)} - 1\right| \]

   15. Applied egg-rr 3.5%

       \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{0.047619047619047616}{\frac{\sqrt{\pi}}{{x}^{7}}}\right)} - 1}\right| \]
   16. Step-by-step derivation

       1. expm1-def 3.8%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.047619047619047616}{\frac{\sqrt{\pi}}{{x}^{7}}}\right)\right)}\right| \]

       2. expm1-log1p 36.7%

          \[\leadsto \left|\color{blue}{\frac{0.047619047619047616}{\frac{\sqrt{\pi}}{{x}^{7}}}}\right| \]

       3. associate-/r/ 36.7%

          \[\leadsto \left|\color{blue}{\frac{0.047619047619047616}{\sqrt{\pi}} \cdot {x}^{7}}\right| \]

       4. *-commutative 36.7%

          \[\leadsto \left|\color{blue}{{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}}\right| \]

   17. Simplified 36.7%

       \[\leadsto \left|\color{blue}{{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 5: 67.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (fabs (* x (/ 2.0 (sqrt PI))))
   (fabs (* (pow x 7.0) (/ 0.047619047619047616 (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((pow(x, 7.0) * (0.047619047619047616 / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs((Math.pow(x, 7.0) * (0.047619047619047616 / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs((math.pow(x, 7.0) * (0.047619047619047616 / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(Float64((x ^ 7.0) * Float64(0.047619047619047616 / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = abs((x * (2.0 / sqrt(pi))));
	else
		tmp = abs(((x ^ 7.0) * (0.047619047619047616 / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Power[x, 7.0], $MachinePrecision] * N[(0.047619047619047616 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left|\frac{\left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot 0.6666666666666666, x, 2\right) + \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{4}, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6}\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around 0 99.0%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right)\right)}\right| \]
   5. Step-by-step derivation

      1. *-commutative 99.0%

         \[\leadsto \left|\color{blue}{\left(\left|x\right| \cdot \left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right)\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

      2. associate-*l* 99.1%

         \[\leadsto \left|\color{blue}{\left|x\right| \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

      3. unpow1 99.1%

         \[\leadsto \left|\left|\color{blue}{{x}^{1}}\right| \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      4. sqr-pow 30.8%

         \[\leadsto \left|\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      5. fabs-sqr 30.8%

         \[\leadsto \left|\color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      6. sqr-pow 99.1%

         \[\leadsto \left|\color{blue}{{x}^{1}} \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      7. unpow1 99.1%

         \[\leadsto \left|\color{blue}{x} \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

   6. Simplified 99.1%

      \[\leadsto \left|\color{blue}{x \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

   7. Taylor expanded in x around 0 68.1%

      \[\leadsto \left|\color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   8. Step-by-step derivation

      1. *-commutative 68.1%

         \[\leadsto \left|\color{blue}{\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot 2}\right| \]

      2. associate-*l* 68.1%

         \[\leadsto \left|\color{blue}{x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 2\right)}\right| \]

      3. *-commutative 68.1%

         \[\leadsto \left|x \cdot \color{blue}{\left(2 \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

   9. Simplified 68.1%

      \[\leadsto \left|\color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   10. Step-by-step derivation

       1. add-log-exp 34.8%

          \[\leadsto \left|\color{blue}{\log \left(e^{x \cdot \left(2 \cdot \sqrt{\frac{1}{\pi}}\right)}\right)}\right| \]

       2. *-commutative 34.8%

          \[\leadsto \left|\log \left(e^{\color{blue}{\left(2 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot x}}\right)\right| \]

       3. exp-prod 34.8%

          \[\leadsto \left|\log \color{blue}{\left({\left(e^{2 \cdot \sqrt{\frac{1}{\pi}}}\right)}^{x}\right)}\right| \]

       4. sqrt-div 34.8%

          \[\leadsto \left|\log \left({\left(e^{2 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}}\right)}^{x}\right)\right| \]

       5. metadata-eval 34.8%

          \[\leadsto \left|\log \left({\left(e^{2 \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}}\right)}^{x}\right)\right| \]

       6. un-div-inv 34.8%

          \[\leadsto \left|\log \left({\left(e^{\color{blue}{\frac{2}{\sqrt{\pi}}}}\right)}^{x}\right)\right| \]

   11. Applied egg-rr 34.8%

       \[\leadsto \left|\color{blue}{\log \left({\left(e^{\frac{2}{\sqrt{\pi}}}\right)}^{x}\right)}\right| \]
   12. Step-by-step derivation

       1. log-pow 68.1%

          \[\leadsto \left|\color{blue}{x \cdot \log \left(e^{\frac{2}{\sqrt{\pi}}}\right)}\right| \]

       2. rem-log-exp 68.1%

          \[\leadsto \left|x \cdot \color{blue}{\frac{2}{\sqrt{\pi}}}\right| \]

   13. Simplified 68.1%

       \[\leadsto \left|\color{blue}{x \cdot \frac{2}{\sqrt{\pi}}}\right| \]

   if 1.8500000000000001 < x

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left|\frac{\left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot 0.6666666666666666, x, 2\right) + \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{4}, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6}\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around 0 99.0%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right)\right)}\right| \]
   5. Step-by-step derivation

      1. *-commutative 99.0%

         \[\leadsto \left|\color{blue}{\left(\left|x\right| \cdot \left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right)\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

      2. associate-*l* 99.1%

         \[\leadsto \left|\color{blue}{\left|x\right| \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

      3. unpow1 99.1%

         \[\leadsto \left|\left|\color{blue}{{x}^{1}}\right| \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      4. sqr-pow 30.8%

         \[\leadsto \left|\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      5. fabs-sqr 30.8%

         \[\leadsto \left|\color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      6. sqr-pow 99.1%

         \[\leadsto \left|\color{blue}{{x}^{1}} \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      7. unpow1 99.1%

         \[\leadsto \left|\color{blue}{x} \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

   6. Simplified 99.1%

      \[\leadsto \left|\color{blue}{x \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

   7. Taylor expanded in x around inf 36.6%

      \[\leadsto \left|x \cdot \color{blue}{\left(0.047619047619047616 \cdot \left({x}^{6} \cdot \sqrt{\frac{1}{\pi}}\right)\right)}\right| \]
   8. Step-by-step derivation

      1. associate-*r* 36.7%

         \[\leadsto \left|x \cdot \color{blue}{\left(\left(0.047619047619047616 \cdot {x}^{6}\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

   9. Simplified 36.7%

      \[\leadsto \left|x \cdot \color{blue}{\left(\left(0.047619047619047616 \cdot {x}^{6}\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   10. Step-by-step derivation

       1. expm1-log1p-u 3.8%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(\left(0.047619047619047616 \cdot {x}^{6}\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)}\right| \]

       2. expm1-udef 3.5%

          \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(x \cdot \left(\left(0.047619047619047616 \cdot {x}^{6}\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)} - 1}\right| \]

       3. associate-*l* 3.5%

          \[\leadsto \left|e^{\mathsf{log1p}\left(x \cdot \color{blue}{\left(0.047619047619047616 \cdot \left({x}^{6} \cdot \sqrt{\frac{1}{\pi}}\right)\right)}\right)} - 1\right| \]

       4. sqrt-div 3.5%

          \[\leadsto \left|e^{\mathsf{log1p}\left(x \cdot \left(0.047619047619047616 \cdot \left({x}^{6} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right)\right)\right)} - 1\right| \]

       5. metadata-eval 3.5%

          \[\leadsto \left|e^{\mathsf{log1p}\left(x \cdot \left(0.047619047619047616 \cdot \left({x}^{6} \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right)\right)\right)} - 1\right| \]

       6. un-div-inv 3.5%

          \[\leadsto \left|e^{\mathsf{log1p}\left(x \cdot \left(0.047619047619047616 \cdot \color{blue}{\frac{{x}^{6}}{\sqrt{\pi}}}\right)\right)} - 1\right| \]

   11. Applied egg-rr 3.5%

       \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(x \cdot \left(0.047619047619047616 \cdot \frac{{x}^{6}}{\sqrt{\pi}}\right)\right)} - 1}\right| \]
   12. Step-by-step derivation

       1. expm1-def 3.8%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(0.047619047619047616 \cdot \frac{{x}^{6}}{\sqrt{\pi}}\right)\right)\right)}\right| \]

       2. expm1-log1p 36.7%

          \[\leadsto \left|\color{blue}{x \cdot \left(0.047619047619047616 \cdot \frac{{x}^{6}}{\sqrt{\pi}}\right)}\right| \]

       3. associate-*r/ 36.7%

          \[\leadsto \left|x \cdot \color{blue}{\frac{0.047619047619047616 \cdot {x}^{6}}{\sqrt{\pi}}}\right| \]

       4. associate-*r/ 36.7%

          \[\leadsto \left|\color{blue}{\frac{x \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}}\right| \]

       5. *-commutative 36.7%

          \[\leadsto \left|\frac{\color{blue}{\left(0.047619047619047616 \cdot {x}^{6}\right) \cdot x}}{\sqrt{\pi}}\right| \]

       6. associate-*l* 36.7%

          \[\leadsto \left|\frac{\color{blue}{0.047619047619047616 \cdot \left({x}^{6} \cdot x\right)}}{\sqrt{\pi}}\right| \]

       7. pow-plus 36.7%

          \[\leadsto \left|\frac{0.047619047619047616 \cdot \color{blue}{{x}^{\left(6 + 1\right)}}}{\sqrt{\pi}}\right| \]

       8. metadata-eval 36.7%

          \[\leadsto \left|\frac{0.047619047619047616 \cdot {x}^{\color{blue}{7}}}{\sqrt{\pi}}\right| \]

   13. Simplified 36.7%

       \[\leadsto \left|\color{blue}{\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}}\right| \]
   14. Step-by-step derivation

       1. expm1-log1p-u 3.8%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right)\right)}\right| \]

       2. expm1-udef 3.5%

          \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right)} - 1}\right| \]

       3. associate-/l* 3.5%

          \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{0.047619047619047616}{\frac{\sqrt{\pi}}{{x}^{7}}}}\right)} - 1\right| \]

   15. Applied egg-rr 3.5%

       \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{0.047619047619047616}{\frac{\sqrt{\pi}}{{x}^{7}}}\right)} - 1}\right| \]
   16. Step-by-step derivation

       1. expm1-def 3.8%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.047619047619047616}{\frac{\sqrt{\pi}}{{x}^{7}}}\right)\right)}\right| \]

       2. expm1-log1p 36.7%

          \[\leadsto \left|\color{blue}{\frac{0.047619047619047616}{\frac{\sqrt{\pi}}{{x}^{7}}}}\right| \]

       3. associate-/r/ 36.7%

          \[\leadsto \left|\color{blue}{\frac{0.047619047619047616}{\sqrt{\pi}} \cdot {x}^{7}}\right| \]

       4. *-commutative 36.7%

          \[\leadsto \left|\color{blue}{{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}}\right| \]

   17. Simplified 36.7%

       \[\leadsto \left|\color{blue}{{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 6: 67.8% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.6666666666666666 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.7)
   (fabs (* x (/ 2.0 (sqrt PI))))
   (fabs (* 0.6666666666666666 (* (sqrt (/ 1.0 PI)) (* x (* x x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.7) {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((0.6666666666666666 * (sqrt((1.0 / ((double) M_PI))) * (x * (x * x)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.7) {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs((0.6666666666666666 * (Math.sqrt((1.0 / Math.PI)) * (x * (x * x)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.7:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs((0.6666666666666666 * (math.sqrt((1.0 / math.pi)) * (x * (x * x)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.7)
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(Float64(0.6666666666666666 * Float64(sqrt(Float64(1.0 / pi)) * Float64(x * Float64(x * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.7)
		tmp = abs((x * (2.0 / sqrt(pi))));
	else
		tmp = abs((0.6666666666666666 * (sqrt((1.0 / pi)) * (x * (x * x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.7], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.6666666666666666 * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.69999999999999996

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left|\frac{\left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot 0.6666666666666666, x, 2\right) + \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{4}, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6}\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around 0 99.0%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right)\right)}\right| \]
   5. Step-by-step derivation

      1. *-commutative 99.0%

         \[\leadsto \left|\color{blue}{\left(\left|x\right| \cdot \left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right)\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

      2. associate-*l* 99.1%

         \[\leadsto \left|\color{blue}{\left|x\right| \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

      3. unpow1 99.1%

         \[\leadsto \left|\left|\color{blue}{{x}^{1}}\right| \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      4. sqr-pow 30.8%

         \[\leadsto \left|\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      5. fabs-sqr 30.8%

         \[\leadsto \left|\color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      6. sqr-pow 99.1%

         \[\leadsto \left|\color{blue}{{x}^{1}} \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      7. unpow1 99.1%

         \[\leadsto \left|\color{blue}{x} \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

   6. Simplified 99.1%

      \[\leadsto \left|\color{blue}{x \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

   7. Taylor expanded in x around 0 68.1%

      \[\leadsto \left|\color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   8. Step-by-step derivation

      1. *-commutative 68.1%

         \[\leadsto \left|\color{blue}{\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot 2}\right| \]

      2. associate-*l* 68.1%

         \[\leadsto \left|\color{blue}{x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 2\right)}\right| \]

      3. *-commutative 68.1%

         \[\leadsto \left|x \cdot \color{blue}{\left(2 \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

   9. Simplified 68.1%

      \[\leadsto \left|\color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   10. Step-by-step derivation

       1. add-log-exp 34.8%

          \[\leadsto \left|\color{blue}{\log \left(e^{x \cdot \left(2 \cdot \sqrt{\frac{1}{\pi}}\right)}\right)}\right| \]

       2. *-commutative 34.8%

          \[\leadsto \left|\log \left(e^{\color{blue}{\left(2 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot x}}\right)\right| \]

       3. exp-prod 34.8%

          \[\leadsto \left|\log \color{blue}{\left({\left(e^{2 \cdot \sqrt{\frac{1}{\pi}}}\right)}^{x}\right)}\right| \]

       4. sqrt-div 34.8%

          \[\leadsto \left|\log \left({\left(e^{2 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}}\right)}^{x}\right)\right| \]

       5. metadata-eval 34.8%

          \[\leadsto \left|\log \left({\left(e^{2 \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}}\right)}^{x}\right)\right| \]

       6. un-div-inv 34.8%

          \[\leadsto \left|\log \left({\left(e^{\color{blue}{\frac{2}{\sqrt{\pi}}}}\right)}^{x}\right)\right| \]

   11. Applied egg-rr 34.8%

       \[\leadsto \left|\color{blue}{\log \left({\left(e^{\frac{2}{\sqrt{\pi}}}\right)}^{x}\right)}\right| \]
   12. Step-by-step derivation

       1. log-pow 68.1%

          \[\leadsto \left|\color{blue}{x \cdot \log \left(e^{\frac{2}{\sqrt{\pi}}}\right)}\right| \]

       2. rem-log-exp 68.1%

          \[\leadsto \left|x \cdot \color{blue}{\frac{2}{\sqrt{\pi}}}\right| \]

   13. Simplified 68.1%

       \[\leadsto \left|\color{blue}{x \cdot \frac{2}{\sqrt{\pi}}}\right| \]

   if 1.69999999999999996 < x

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left|\frac{\left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot 0.6666666666666666, x, 2\right) + \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{4}, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6}\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around inf 28.3%

      \[\leadsto \left|\color{blue}{0.6666666666666666 \cdot \left(\left({x}^{2} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   5. Step-by-step derivation

      1. *-commutative 28.3%

         \[\leadsto \left|0.6666666666666666 \cdot \left(\color{blue}{\left(\left|x\right| \cdot {x}^{2}\right)} \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      2. *-commutative 28.3%

         \[\leadsto \left|0.6666666666666666 \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot {x}^{2}\right)\right)}\right| \]

      3. unpow2 28.3%

         \[\leadsto \left|0.6666666666666666 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right| \]

      4. sqr-abs 28.3%

         \[\leadsto \left|0.6666666666666666 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)}\right)\right)\right| \]

      5. cube-mult 28.3%

         \[\leadsto \left|0.6666666666666666 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \color{blue}{{\left(\left|x\right|\right)}^{3}}\right)\right| \]

      6. unpow1 28.3%

         \[\leadsto \left|0.6666666666666666 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {\left(\left|\color{blue}{{x}^{1}}\right|\right)}^{3}\right)\right| \]

      7. sqr-pow 2.1%

         \[\leadsto \left|0.6666666666666666 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {\left(\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|\right)}^{3}\right)\right| \]

      8. fabs-sqr 2.1%

         \[\leadsto \left|0.6666666666666666 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {\color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}}^{3}\right)\right| \]

      9. sqr-pow 28.3%

         \[\leadsto \left|0.6666666666666666 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {\color{blue}{\left({x}^{1}\right)}}^{3}\right)\right| \]

      10. unpow1 28.3%

          \[\leadsto \left|0.6666666666666666 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {\color{blue}{x}}^{3}\right)\right| \]

   6. Simplified 28.3%

      \[\leadsto \left|\color{blue}{0.6666666666666666 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{3}\right)}\right| \]
   7. Step-by-step derivation

      1. unpow3 28.3%

         \[\leadsto \left|0.6666666666666666 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}\right)\right| \]

   8. Applied egg-rr 28.3%

      \[\leadsto \left|0.6666666666666666 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}\right)\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.6666666666666666 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right|\\ \end{array} \]

Alternative 7: 67.8% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \left|x \cdot \frac{2}{\sqrt{\pi}}\right| \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fabs (* x (/ 2.0 (sqrt PI)))))

C

double code(double x) {
	return fabs((x * (2.0 / sqrt(((double) M_PI)))));
}

Java

public static double code(double x) {
	return Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
}

Python

def code(x):
	return math.fabs((x * (2.0 / math.sqrt(math.pi))))

Julia

function code(x)
	return abs(Float64(x * Float64(2.0 / sqrt(pi))))
end

MATLAB

function tmp = code(x)
	tmp = abs((x * (2.0 / sqrt(pi))));
end

Wolfram

code[x_] := N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{\left|\frac{\left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot 0.6666666666666666, x, 2\right) + \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{4}, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6}\right)\right)}{\sqrt{\pi}}\right|} \]

4. Taylor expanded in x around 0 99.0%

   \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right)\right)}\right| \]
5. Step-by-step derivation

   1. *-commutative 99.0%

      \[\leadsto \left|\color{blue}{\left(\left|x\right| \cdot \left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right)\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

   2. associate-*l* 99.1%

      \[\leadsto \left|\color{blue}{\left|x\right| \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

   3. unpow1 99.1%

      \[\leadsto \left|\left|\color{blue}{{x}^{1}}\right| \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

   4. sqr-pow 30.8%

      \[\leadsto \left|\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

   5. fabs-sqr 30.8%

      \[\leadsto \left|\color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

   6. sqr-pow 99.1%

      \[\leadsto \left|\color{blue}{{x}^{1}} \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

   7. unpow1 99.1%

      \[\leadsto \left|\color{blue}{x} \cdot \left(\left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

6. Simplified 99.1%

   \[\leadsto \left|\color{blue}{x \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

7. Taylor expanded in x around 0 68.1%

   \[\leadsto \left|\color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
8. Step-by-step derivation

   1. *-commutative 68.1%

      \[\leadsto \left|\color{blue}{\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot 2}\right| \]

   2. associate-*l* 68.1%

      \[\leadsto \left|\color{blue}{x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 2\right)}\right| \]

   3. *-commutative 68.1%

      \[\leadsto \left|x \cdot \color{blue}{\left(2 \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

9. Simplified 68.1%

   \[\leadsto \left|\color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
10. Step-by-step derivation

    1. add-log-exp 34.8%

       \[\leadsto \left|\color{blue}{\log \left(e^{x \cdot \left(2 \cdot \sqrt{\frac{1}{\pi}}\right)}\right)}\right| \]

    2. *-commutative 34.8%

       \[\leadsto \left|\log \left(e^{\color{blue}{\left(2 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot x}}\right)\right| \]

    3. exp-prod 34.8%

       \[\leadsto \left|\log \color{blue}{\left({\left(e^{2 \cdot \sqrt{\frac{1}{\pi}}}\right)}^{x}\right)}\right| \]

    4. sqrt-div 34.8%

       \[\leadsto \left|\log \left({\left(e^{2 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}}\right)}^{x}\right)\right| \]

    5. metadata-eval 34.8%

       \[\leadsto \left|\log \left({\left(e^{2 \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}}\right)}^{x}\right)\right| \]

    6. un-div-inv 34.8%

       \[\leadsto \left|\log \left({\left(e^{\color{blue}{\frac{2}{\sqrt{\pi}}}}\right)}^{x}\right)\right| \]

11. Applied egg-rr 34.8%

    \[\leadsto \left|\color{blue}{\log \left({\left(e^{\frac{2}{\sqrt{\pi}}}\right)}^{x}\right)}\right| \]
12. Step-by-step derivation

    1. log-pow 68.1%

       \[\leadsto \left|\color{blue}{x \cdot \log \left(e^{\frac{2}{\sqrt{\pi}}}\right)}\right| \]

    2. rem-log-exp 68.1%

       \[\leadsto \left|x \cdot \color{blue}{\frac{2}{\sqrt{\pi}}}\right| \]

13. Simplified 68.1%

    \[\leadsto \left|\color{blue}{x \cdot \frac{2}{\sqrt{\pi}}}\right| \]

14. Final simplification 68.1%

    \[\leadsto \left|x \cdot \frac{2}{\sqrt{\pi}}\right| \]

Reproduce

herbie shell --seed 2023271 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  :precision binary64
  :pre (<= x 0.5)
  (fabs (* (/ 1.0 (sqrt PI)) (+ (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1.0 5.0) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (/ 1.0 21.0) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)))))))